A regularisation of the Coleff-Herrera residue current
The aim of this note is to outline a proof of the following result. Let \(X\) be a complex manifold and let \(f = (f_1,f_2): X\to\mathbb C^2\) be a holomorphic mapping. Assume that \(f\) defines a complete intersection in \(X\), i.e. that \(V_f=\{f_1=f_2 = 0\}\) has codimension 2 in \(X\). Then \[ \lim_{\varepsilon_1,\varepsilon_2\to0}\int{\overline\partial}\,\frac{\overline f_1}{| f_1| ^2+\varepsilon_1}\wedge {\overline\partial}\,\frac{\overline f_2}{| f_2| ^2+ \varepsilon_2}\wedge \varphi=\left[{\overline\partial}\,\frac{1}{f_1}\wedge {\overline\partial}\,\frac{1}{f_2}\right]\cdot\varphi \] for all test forms \(\varphi\) of bidegree \((n,n-2)\). Hence, the Coleff-Herrera current can be obtained as the unrestricted (weak) limit of a smooth \((0,2)\)-form depending on two parameters.
scientific article; zbMATH DE number 2118306